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The ratio, mean-of-the ratios and Horvitz-Thompson estimators under the continuous variable model Chamwali, Anthony Alifa

Abstract

This study investigates the performances of the ratio estimator, the mean-of-the-ratios estimator and the Horvitz-Thompson (HT) estimator under the continuous variable model of Cassel and Sarndal (1972a, 1972b, 1973). Under this model, the character, Y, which is of interest to the investigator is assumed to be related to an auxiliary variable, X, by Y(Xi) = θ(Xi + Z(Xi)) where ℇ(Zi | Xi) = 0; ∀Xi ℇ (0, ∞); ℇ(Zi² | Xi) = σ² (Xi) = k² Xi[sup g]; ℇ(ZiZj | XiXj) =0; (i ≠ j). It is assumed, in this paper, that X is gamma distributed over (0, ∞) with parameter r. The mean of Y is to be estimated, under the additional assumptions that the design function, P(x), is l) polynominal 2) exponential, i.e. [formulas are not included]. It is observed that for g = 0 or 1, the ratio estimator performs better than the other two. For g = 0, 1 or 2, and for a wider range of values of m or c, the mean-of-the-ratios estimator performs better than the HT estimator. When P(X) is polynominal, the III estimator is most efficient if the sampling design is approximately pps. The results compare well with those of other researchers under similar assumptions.

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